Griess algebra
Encyclopedia
In mathematics
, the Griess algebra is a commutative non-associative algebra on a real
vector space
of dimension
196884 that has the Monster group
M as its automorphism group. It is named after mathematician R. L. Griess
, who constructed it in 1980 and subsequently used it in 1982 to construct M. The Monster fixes (vectorwise) a 1-space in this algebra and acts absolutely irreducibly on the 196883-dimensional orthogonal complement of this 1-space.
(The Monster preserves the standard inner product on the 196884-space.)
Griess's construction was later simplified by Jacques Tits
and John H. Conway.
The Griess algebra is the same as the degree 2 piece of the monster vertex algebra
, and the Griess product is one of the vertex algebra products.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Griess algebra is a commutative non-associative algebra on a real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
of dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
196884 that has the Monster group
Monster group
In the mathematical field of group theory, the Monster group M or F1 is a group of finite order:...
M as its automorphism group. It is named after mathematician R. L. Griess
R. L. Griess
Robert Louis Griess, Jr. is a mathematician working on finite simple groups and vertex algebras. He constructed the monster group using the Griess algebra. He received the AMS Leroy P...
, who constructed it in 1980 and subsequently used it in 1982 to construct M. The Monster fixes (vectorwise) a 1-space in this algebra and acts absolutely irreducibly on the 196883-dimensional orthogonal complement of this 1-space.
(The Monster preserves the standard inner product on the 196884-space.)
Griess's construction was later simplified by Jacques Tits
Jacques Tits
Jacques Tits is a Belgian and French mathematician who works on group theory and geometry and who introduced Tits buildings, the Tits alternative, and the Tits group.- Career :Tits received his doctorate in mathematics at the age of 20...
and John H. Conway.
The Griess algebra is the same as the degree 2 piece of the monster vertex algebra
Monster vertex algebra
The monster vertex algebra is a vertex algebra acted on by the monster group that was constructed by Igor Frenkel, James Lepowsky, and Arne Meurman. R...
, and the Griess product is one of the vertex algebra products.